9. PLANE GEOMETRY PLANE FIGURES Angle x at vertex C is the same as angle x at vertex A (alternate angles). In mathematics, a plane is a flat or two-dimensional At the vertex A, x + y + z = 1800 (angles on a straight surface that has no thickness that and so the term line). ‘plane figures’ is used to describe figures that are But, x, y and z are the interior angles of the triangle drawn on a plane. Circles, ellipses, triangles, ABC. Hence, the sum of the interior angles of a quadrilaterals and other polygons are some examples triangle must be 1800. of plane figures. All plane figures are two- dimensional in nature and the study of these shapes is known as plane geometry or Euclidean geometry. Triangles A triangle is a plane figure bounded by three straight lines. It has three sides and three angles. Hence, there We can now state a general rule connecting the three are six elements in a triangle that can be measured. interior angles of a triangle. The sides meet at three points called the vertices (singular vertex). There are many different types of The sum of the interior angles of a triangle is triangles and although all triangles have some 1800. common properties, different types have their own special additional properties as well. Example 1 In the triangle, ABC, Bˆ =°32 and Cˆ =°48 ; Naming triangles ˆ calculate the size ofA . A triangle is named using the three letters that refer to their vertices. Vertices are named using ‘upper case’ letters and sides opposite the vertex are named using the corresponding lower case letters. Solution Aˆ +°+°=°32 48 180 \Aˆ =180 °-() 32°+° 48 =°100 (The sum of the interior angles in a triangle =180º) In the triangle, the vertices are A, B and C, and so, the triangle is named ABC. The side opposite vertex A, BC is denoted a, the side Exterior angle of a triangle opposite vertexwww.faspassmaths.com B is denoted b and the side opposite angle C is named c. If we extend the side of a triangle, the angle created between the extended side and the ‘next’ side of the triangle is called an exterior angle. Sum of the angles in a triangle A triangle has 3 interior angles and three exterior angles. The exterior angles are formed by extending The three angles at the vertices of a triangle are called each side, moving either in a clockwise direction or interior angles. In the diagram below, x, y and z are in an anticlockwise direction. Exterior angles are the interior angles of the triangle. shown below. Angle y at vertex B is the same as angle y at vertex A (alternate angles). 62 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com Example 2 In PQR, angle PQR= 45°, and angle QPR = 72°. Calculate the measure of the measure of the exterior angle at R. P 720 450 Q R S Exterior Angle Theorem Solution In the triangle below, d is an exterior angle and a, b The exterior angle at R, angle PRS, is equal to the and c, are interior angles. We refer to the angles a sum of the two interior angles at R and at Q, and b as the interior opposite angles since they are opposite to the exterior angle, d. PRS = 450 + 720 PRS = 1170 General properties of triangles The following relationships are true for all triangles, regardless of the shape or size. 1. The shortest side is always opposite the smallest interior angle. 2. The longest side is always opposite the largest interior angle. We already know that the sum of the interior angles 3. If two sides are equal, then the equal sides in a triangle is 180°. That is, are opposite equal angles. 4. If all sides are equal, then all three angles � + � + � = 180° have the same measure of 600. , Also, � + � = 180 (d and c lie on a straight 5. The sum of the lengths of any two sides of a line) triangle is greater than the length of the third Therefore, � + � + � = � + � = 1800 side (triangle inequality theorem). Or (� + �) + � = � + � Subtracting c from each side of the equation, we have Types of triangles (� + �) = � In describing types of triangles, we should note that there are two classifications. We can classify We can now state a general rule connecting the www.faspassmaths.comtriangles by: exterior of a triangle and the interior opposite angles of the triangle. (a) the relationship between the length of their sides (b) the relationship between the measure of their Exterior Angle Theorem: angles The measure of the exterior angle of a triangle is As we shall soon see, these classifications are not equal to the sum of the measures of the two interior mutually exclusive as it is possible to have a triangle opposite angles. that belongs to both sets. 63 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com Classification of triangles by their sides Scalene Triangle Isosceles triangle Two No sides of the same sides are the same length length No equal angles Angles opposite equal sides are equal Equilateral triangle Example 3 All sides are the same length In the figure, triangle MNP is isosceles with NP= NM . If Mˆ =°35 , calculate the measure of Pˆ andNˆ . All angles are equal to 600 Solution Classification of triangles by their angles Pˆ =°35 (The base angles of an isosceles triangle Acute-angled Obtuse-angled triangle are equal) triangle Nˆ =°180-() 35°+° 35 =°110 (The sum of the angles in a triangle is 180º) One angle is obtuse Congruence All angles are acute Two plane shapes are said to be congruent if they Right-angled triangle have the exact same shape and size. Hence, if either one is placed atop the other, there is no overlap. In other words, congruent shapes are the same in all respects and either one can replace the other without detection. www.faspassmaths.comCongruent triangles One angle is right Before we introduce the idea of congruence in triangles, it is important to understand certain key terms. We used the term corresponding angles with As mentioned above, a given triangle can have reference to parallel lines earlier in this chapter. In properties that come from both sets. For example, the same way, we can refer to corresponding angles consider the following scalene triangles in the or even corresponding sides with reference to diagram below. each one is different in terms of the triangles. If the congruent triangles, shown below, are type of angles. superimposed, corresponding sides will coincide with each other and corresponding angles will 64 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com coincide. For example, in the diagram below, the side 2. Two sides and the included angle (SAS) AB will coincide with PQ and so we say that AB If two pairs of sides of two triangles are equal in corresponds to PQ. So too, the angle at B will length, and the measures of the included angles coincide with the angle at Q and we say that angle B are equal, then the triangles are congruent. corresponds to angle Q and so on. Corresponding angles Corresponding sides In triangle ABC and PQR ∠� and ∠� AB and PQ 1. �� = �� ∠� and ∠� AC and PR 2. �� = �� ∠� and ∠� BC and QR 3. ∠��� = ∠���. Therefore, Δ��� ≅ Δ���. Conditions for congruence 3. Two angles and the corresponding side (ASA). When two triangles are congruent, the three If two pairs of angles of two triangles are equal corresponding sides and the three corresponding in measurement, and the included sides are equal angles are equal. However, if we wish to prove that in length, then the triangles are congruent. two triangles are congruent, it is not necessary to prove that all six elements are identical. We can apply certain tests for congruency and this will allow us to determine very easily if two triangles are congruent. These tests outline a set of minimum conditions that must be satisfied for congruency to be concluded. There are four different tests named below. Each test In triangle ABC and PQR comprises a set of three conditions, which determine 1. �� = �� whether or not a pair of triangles are congruent. If 2. ∠��� = ∠��� any one of these tests is satisfied then the triangles 3. ∠��� = ∠���. are congruent. These tests are described below. Therefore, Δ��� ≅ Δ���. 1. Three sides (SSS) 4. Right angle, hypotenuse and side (RHS). If three pairs of sides of two triangles are equal in If two right-angled triangles have their hypotenuses length, then the triangles are congruent. equal in length, and a pair of shorter sides are also equal in length, then the triangles are congruent. www.faspassmaths.com In triangle ABC and PQR 1. �� = �� In triangle ABC and PQR 2. �� = �� 1. ∠��� = ∠��� = 90, 3. �� = ��. 2. �� = �� Therefore, Δ��� ≅ Δ���. 3. �� = ��. The symbol ≅ means ‘is congruent to’. Therefore, Δ��� ≅ Δ���. 65 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com Example 4 State, with reason, whether the triangles shown If we reversed the order of comparison, then we below are congruent. would have �� �� �� 1 = = = �� �� �� � In stating the ratios, one must be consistent in maintaining the order. Notice that the above ratio compares the sides of the smaller triangle with the sides of the larger triangle. Similar Figures Solution If any two plane figures are similar, then In triangles ACB and XYZ: i. the figures are equiangular 1. ∠ACB=∠XZY ii. the ratios of their corresponding sides are 2. ∠CAB=∠XYZ the same. 3. AB=XY The converse is also true, so if we know that a pair Therefore, triangles ACB and XYZ are congruent of figures have corresponding sides in a fixed ratio, (two angles and the corresponding side).